Subgroups — #490653

Order 512.

407 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33734	C₂	1	1	yes	maximal, characteristic, complemented
2	#33480	C₂	1	1	yes	maximal, characteristic
3	#19223	C₂	1	2	yes	maximal
4	#12482	C₂	1	2	yes	maximal, complemented
5	#12474	C₂	1	2	yes	maximal, complemented
6	#12457	C₂	1	1	yes	maximal, characteristic, complemented
7	#12451	C₂	1	1	yes	maximal, characteristic, complemented
8	#12440	C₂	1	1	yes	maximal, characteristic, complemented
9	#12431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#1371	C₂	1	1	yes	maximal, characteristic, complemented
11	#11954	C₂	1	2	yes	maximal, complemented
12	#3248	—	2	2	no	—
13	#3170	C₂²	1	2	yes	—
14	#2966	C₂²	1	1	yes	characteristic
15	#2252	C₂²	1	1	yes	characteristic
16	#2247	C₂²	1	1	yes	characteristic
17	#2241	C₂²	1	2	yes	—
18	#2216	—	2	1	no	—
19	#2216	—	2	1	no	—
20	#2183	C₂²	1	1	yes	characteristic
21	Q₈×C₁₆	C₂²	1	1	yes	characteristic
22	C₄×Q₃₂	—	2	1	no	—
23	C₄×Q₃₂	—	2	1	no	—
24	#2169	C₂²	1	1	yes	characteristic
25	#2168	—	2	1	no	—
26	D₄×C₁₆	—	2	1	no	—
27	D₄×C₁₆	C₂²	1	1	yes	characteristic
28	#2164	—	2	1	no	—
29	#2163	C₂²	1	1	yes	characteristic
30	#2108	—	2	1	no	—
31	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	#1700	—	2	2	no	—
33	#1699	—	2	2	no	—
34	#1674	C₂²	1	2	yes	—
35	#1673	C₂²	1	2	yes	—
36	#1670	C₂²	1	2	yes	—
37	#1669	C₂²	1	2	yes	—
38	C₂×#335	C₂²	1	2	yes	—
39	#1552	—	2	2	no	—
40	#1371	C₂²	1	1	yes	characteristic
41	#1371	C₂²	1	1	yes	characteristic
42	#1371	C₂²	1	2	yes	—
43	#1364	C₂²	1	2	yes	—
44	#1364	C₂²	1	2	yes	—
45	#1362	C₂²	1	2	yes	—
46	#1362	C₂²	1	2	yes	—
47	#533	—	2	2	no	—
48	#529	D₄	1	1	yes	characteristic
49	C₂×Q₃₂	—	2	1	no	—
50	C₂×Q₃₂	—	2	1	no	—
51	C₂×M₃₂	—	2	1	no	—
52	C₂²×C₁₆	—	2	1	no	—
53	C₂²×C₁₆	D₄	1	1	yes	characteristic
54	C₂²×C₁₆	D₄	1	1	yes	characteristic
55	#500	C₂³	1	2	yes	—
56	#494	C₂³	1	2	yes	—
57	#487	—	2	2	no	—
58	#479	—	4	2	no	—
59	#475	—	2	2	no	—
60	#474	—	2	2	no	—
61	#467	—	2	2	no	—
62	#467	—	2	2	no	—
63	#467	—	4	2	no	—
64	#465	—	4	2	no	—
65	#464	—	4	2	no	—
66	#462	—	2	2	no	—
67	#461	—	2	2	no	—
68	#446	D₄	1	1	yes	characteristic
69	C₈×Q₈	C₂³	1	1	yes	characteristic
70	C₄×Q₁₆	—	2	2	no	—
71	C₄×Q₁₆	—	2	2	no	—
72	#436	D₄	1	1	yes	characteristic
73	#435	—	2	1	no	—
74	C₈×D₄	—	2	1	no	—
75	C₈×D₄	C₂³	1	1	yes	characteristic
76	#433	—	2	1	no	—
77	#432	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₈	—	2	2	no	—
79	#405	—	2	1	no	—
80	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
81	#366	—	2	1	no	—
82	#366	—	2	1	no	—
83	C₄⋊C₁₆	—	2	1	no	—
84	C₄⋊C₁₆	—	2	1	no	—
85	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
86	C₄⋊C₁₆	C₂³	1	1	yes	characteristic
87	#358	—	2	2	no	—
88	#358	—	2	2	no	—
89	C₂²⋊C₁₆	—	2	1	no	—
90	C₂²⋊C₁₆	—	2	1	no	—
91	C₂²⋊C₁₆	—	2	1	no	—
92	#346	—	2	1	no	—
93	#346	—	2	1	no	—
94	C₄×C₁₆	—	2	1	no	—
95	C₄×C₁₆	D₄	1	1	yes	characteristic
96	C₄×C₁₆	D₄	1	1	yes	characteristic
97	C₄×C₁₆	C₂³	1	1	yes	characteristic
98	C₄×C₁₆	C₂³	1	1	yes	characteristic
99	#335	—	2	2	no	—
100	#335	C₂³	1	2	yes	—
101	#335	C₂³	1	2	yes	—
102	#332	—	2	2	no	—
103	#332	—	2	2	no	—
104	#332	—	2	2	no	—
105	#332	—	2	2	no	—
106	C₂×Q₁₆	—	2	2	no	—
107	C₂×Q₁₆	—	2	2	no	—
108	C₂×Q₁₆	—	4	2	no	—
109	C₂×SD₁₆	—	4	2	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	4	2	no	—
113	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₈	#41	1	1	yes	characteristic
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	#126	—	2	1	no	—
118	#125	—	4	2	no	—
119	#125	D₈	1	1	yes	characteristic
120	#124	D₈	1	1	yes	characteristic
121	#123	—	2	1	no	—
122	C₂²⋊Q₈	—	4	2	no	—
123	C₂²⋊Q₈	—	4	2	no	—
124	C₂²⋊Q₈	D₈	1	1	yes	characteristic
125	#121	D₈	1	1	yes	characteristic
126	C₄×Q₈	—	4	2	no	—
127	C₄×Q₈	—	4	2	no	—
128	C₄×Q₈	C₂×D₄	1	1	yes	characteristic
129	C₄×D₄	—	2	1	no	—
130	C₄×D₄	C₂×D₄	1	1	yes	characteristic
131	#117	—	2	1	no	—
132	#117	—	4	2	no	—
133	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
134	Q₃₂	—	4	2	no	—
135	Q₃₂	—	4	2	no	—
136	M₃₂	—	4	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	2	1	no	—
139	C₂×C₁₆	—	2	1	no	—
140	C₂×C₁₆	—	4	1	no	—
141	C₂×C₁₆	—	4	2	no	—
142	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
143	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
144	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
145	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
146	#107	—	2	2	no	—
147	#107	—	2	2	no	—
148	#107	—	4	2	no	—
149	#106	—	4	2	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	4	2	no	—
155	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
156	C₄⋊C₈	C₂×D₄	1	1	yes	characteristic
157	#103	—	4	2	no	—
158	#103	—	4	2	no	—
159	#103	—	4	2	no	—
160	#103	—	4	2	no	—
161	#103	—	4	2	no	—
162	#102	—	4	2	no	—
163	C₂²⋊C₈	—	2	1	no	—
164	C₂²⋊C₈	—	2	1	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	#97	—	2	1	no	—
167	#97	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	C₂×D₄	1	1	yes	characteristic
170	C₄×C₈	C₂×D₄	1	1	yes	characteristic
171	C₄×C₈	#41	1	1	yes	characteristic
172	C₄×C₈	C₂⁴	1	1	yes	characteristic
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	C₂×D₈	1	1	yes	characteristic
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	C₂×D₈	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	#121	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
181	C₂²×C₄	#135	1	1	yes	characteristic
182	Q₁₆	—	4	2	no	—
183	Q₁₆	—	4	2	no	—
184	Q₁₆	—	8	2	no	—
185	Q₁₆	—	16	2	no	—
186	SD₁₆	—	16	2	no	—
187	M₁₆	—	4	2	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	16	2	no	—
198	C₂×C₈	#121	1	1	yes	characteristic
199	C₂×C₈	#121	1	1	yes	characteristic
200	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₈	C₂×#41	1	1	yes	characteristic
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	8	2	no	—
211	C₄⋊C₄	—	8	2	no	—
212	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
213	C₄⋊C₄	C₂×D₈	1	1	yes	characteristic
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	8	2	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	8	2	no	—
222	C₄²	#121	1	1	yes	characteristic
223	C₄²	C₂×D₈	1	1	yes	characteristic
224	C₄²	#135	1	1	yes	characteristic
225	C₄²	C₂²×D₄	1	1	yes	characteristic
226	C₁₆	—	2	1	no	—
227	C₁₆	—	2	1	no	—
228	C₁₆	—	2	1	no	—
229	C₁₆	—	4	1	no	—
230	C₁₆	—	4	1	no	—
231	C₁₆	#135	1	2	yes	—
232	C₂³	—	2	1	no	—
233	C₂³	#466	1	1	yes	characteristic
234	Q₈	—	2	1	no	—
235	Q₈	—	2	1	no	—
236	Q₈	—	8	2	no	—
237	Q₈	—	8	2	no	—
238	Q₈	—	8	2	no	—
239	Q₈	—	8	2	no	—
240	D₄	—	2	1	no	—
241	D₄	—	2	1	no	—
242	D₄	—	4	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	#466	1	1	yes	characteristic
256	C₂×C₄	#466	1	1	yes	characteristic
257	C₂×C₄	#466	1	1	yes	characteristic
258	C₂×C₄	C₂×#121	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
260	C₂×C₄	C₂×#135	1	1	yes	characteristic
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	4	1	no	—
265	C₈	—	4	1	no	—
266	C₈	—	8	2	no	—
267	C₈	—	8	2	no	—
268	C₈	C₂×#135	1	1	yes	characteristic
269	C₈	#575	1	1	yes	characteristic
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	#2222	1	2	yes	—
276	C₂²	C₂×#466	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	16	2	no	—
285	C₄	—	16	2	no	—
286	C₄	#3035	1	1	yes	characteristic
287	C₄	#3067	1	1	yes	characteristic
288	C₄	#3415	1	1	yes	characteristic
289	C₄	#3417	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	8	1	no	—
292	C₂	#19125	1	1	yes	characteristic
293	C₂	#33610	1	1	yes	characteristic
294	C₂	C₂×#2222	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #33734	C₂	semidirect
4. #12482	C₂	semidirect
5. #12474	C₂	semidirect
6. #12457	C₂	semidirect
7. #12451	C₂	semidirect
8. #12440	C₂	semidirect
9. #12431	C₂	semidirect
10. C₂×#1371	C₂	semidirect
11. #11954	C₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#33734	2
#33480	2
#19223	2
#12482	2
#12474	2
#12457	2
#12451	2
#12440	2
#12431	2
C₂×#1371	2
#11954	2